Globally balancing spanning trees

We show that for every graph $G$ that contains two edge-disjoint spanning trees, we can choose two edge-disjoint spanning trees $T_1,T_2$ of $G$ such that $\mathrm{\mid }{d}_{T_1}(v)-d_{T_2}(v)\mathrm{\mid }\le 5$ for all $v\in V(G)$. We also prove the more general statement that for every positive integer $k$, there is a constant $c_k\in O(\log k)$ such that for every graph $G$ that contains $k$ edge-disjoint spanning trees, we can choose $k$ edge-disjoint spanning trees $T_1,\dots ,T_k$ of $G$ satisfying $\mathrm{\mid }{d}_{T_i}(v)-d_{T_j}(v)\mathrm{\mid }\le c_k$ for all $v \in V(G)$and$i,j \in {1,\ldots,k}$. This resolves a conjecture of Kriesell.